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The `frozen' character of general relativity (and background independent theories) is usually considered to be a problem for the theory. However, the most obvious resolution of this problem, involving correlations between dynamical variables, can be shown to provide a natural explanation of the appearance of time in timeless mathematical structures. I argue that this response can resolve a problem with Tegmark's extreme structuralist position, namely how to account for the appearance of time and change given that the structures in question are taken to be mathematical and, therefore, timeless.

Author Bio

Dean Rickles is a lecturer in History and Philosophy of Science at the University of Sydney. He works primarily on the physics of spacetime, but is also engaged in projects relating to public health and financial economics. He has written/edited several books on the philosophy of physics: The Structural Foundations of Quantum Gravity (OUP, 2006 - coedited with S. French and J. Saatsi), Symmetry, Structure and Spacetime (Elsevier, 2007), and The Ashgate Companion to Contemporary Philosophy of Physics (Ashgate, 2008).

I enjoyed reading your essay, particularly the explanation of the relational emergence of time, as well as the presentation of the ultrastructuralist approach. I tried to understand your closing observation:

“if our universe is a mathematical structure, then we have to accept the possibility that there is an identical structure with a radically different appearance. That, as it stands, is very hard to make sense of.”

For an appearance to exist, it has to “appear” to something/somebody, for example to an observer. The “appearance” is, therefore, relational. If the world is a mathematical structure, then the observer, as a part of it, is also relational. The relation is, by definition, preserved by an isomorphism, therefore I don’t understand where the possibility of a radically different appearance resides. In my opinion, if I get it well, to perceive two identical structures as different appearances, the observer should have different relation to the two structures, for example being in this time embedded all three in a larger structure. I can interpret the realization of a structure only as an isomorphic substructure of another structure.

I find appropriate to submit to your interest a slightly complementary structuralist view, which abstracts an essential part of the physical theories, concerning space, time, causality and physical law, in a general mathematical structure (briefly presented in my essay, and more detailed at http://philsci-archive.pitt.edu/archive/00004355/).

Best wishes,

Cristi Stoica

“Flowing with a Frozen River”,

http://fqxi.org/community/forum/topic/322

Dimi Chakalov wrote on Dec. 4, 2008 @ 01:30 GMT

Dean:

May I ask you to help me understand your main idea. You wrote:

" ... general relativity leads us to view spacetime geometry as part of a dynamical system, as something that satisfies equations of motion and evolves. But clearly the evolution here cannot be understood in a temporal sense, unless we have at our disposal some external time parameter in which to understand it."

...

"The observables so 'localized' are relational in the sense that they are not defined on a background space but only relative to other dynamical entities (matter fields, spatial volume, etc.). Observables are not of the form A(x; t) (where x and t label an independent manifold) but A(B) (where B is another observable and neither B nor A is privileged in any sense)."

Footnote 5: "I restrict the discussion to classical systems in order to make the presentation easier to follow. For the technically savvy, one can transform to the quantum case, roughly, by thinking of the functional relation or correlation A(B) as representing the expectation values of A relative to the eigenvalues of B."

I am not "technically savvy" (cf. footnote 5), and cannot grasp the line of thought in the three excerpts from your essay, particularly the adverb "roughly" in footnote 5.

To be specific, the relational emergence of time poses a paradox, which may be explained as follows.

Imagine a herd of Buridan donkeys, with two stacks of hey in front of each donkey, such that the distance from any given donkey to its stacks of hey is determined -- relationally -- by 'the rest of the donkeys in the herd'.

Consider a donkey called A, and denote 'the rest of the donkeys in the herd' with B, to match your idea in the second excerpt above.

We end up with totally halted/frozen set of (Buridan) donkeys, because donkey A has to wait until the distance to its stacks of hey is determined by B , but any donkey that belongs to the subset denoted with B has to wait until the distance to its stacks of hey is determined -- relationally -- by A.

And since none of the donkeys is "privileged in any sense" (cf. above), the same halting occurs for all donkeys.

I restrict the discussion to classical donkeys in order to make the presentation of the paradox easier to follow. Hope you can solve this 'classical Buridan donkey paradox', and show that the "relational emergence of time" matches the time read by your wristwatch. Then please proceed to the mystery outlined in your footnote 5 above.

Good luck.

Dimi Chakalov

Dean Rickles wrote on Dec. 4, 2008 @ 01:58 GMT

Dear Cristi,

You make a good point: this passage wasn't very clearly expressed now I look at it again. The point I was making was simply that a given mathematical structure does not uniquely determine how it is realised or presented. 'Appears' is a badly chosen term. The point was made much more wittily by Hilbert: he noted that some mathematical structure could be given even by suitably interrelated bits of furniture.

I'll take a look at your paper on the philsci archive.

Best,

Dean

Dean Rickles wrote on Dec. 4, 2008 @ 02:16 GMT

Dear Dimi,

The first premise of you paradox can be denied. Nowhere is it stated that the correlations hold between a system and every other system. This procedure is 'local', so that to get the dynamics for the whole universe, you would have to employ different internal times and patch them together (using some suitable transformation rules).

On footnote 5. Take a state representing the spatial geometry (in the loop approach this will be a graph or superposition of such). As explained elsewhere in the essay, physical states must satisfy constraints (now, in the quantum context, understood as operators, where the states must be annihilated by them in order to be 'physically admissible'). Such a physical state will generically represent a superposition of distinct spatial volumes (that is, one has a superposition of distinct volume eigenstates). Now, take a real number b (corresponding to some volume, represent this observable by B). One can project the superposition onto the volume eigenbasis with the eigenvalue b. One can then work out expectation values for other observables A in this projection. Do this for all values b of B and you get an A(B) that encodes the relational evolution.

Best,

Dean

Chris Kennedy wrote on Dec. 10, 2008 @ 01:01 GMT

Hi Don,

Great essay. I must also tell you that I have been a fan of yours for quite some time. Not just the comedy either. I thought your acting in the movie "Casino" with Robert DeNiro and Joe Pesci was terrific. And now you are adding physics to the list. Wonderful. Even as I look at your title page I ...oh, wait....sorry - it's Dean. Of course. Don't I feel like the dummy. Well anyway, I really did enjoy the essay and was particularly thankful that you referenced Einstein's 1918 Twin Paradox paper. I think we may be the only two on this list who have (although for different reasons). By the way - Richard Feynman had a great line comparing Physics and Mathematics but I can't say it here.

Take care,

CJ

Dimi Chakalov wrote on Dec. 10, 2008 @ 14:31 GMT

Dean:

RE your reply from Dec. 4, 2008 @ 02:16 GMT: to get the dynamics for 'the whole universe' -- the only 'truly isolated system' -- your first off task is to define some reference object with respect to which you can talk about 'the whole universe'. As you acknowledged, your procedure is 'local', hence you are forced to "employ different internal times and patch them together (using some suitable transformation rules)", which in turn makes your "relational evolution" look like pulling yourself (and your horse) out of the swamp by your own hair (Baron Munchausen).

The task is known since Aristotle -- recall The First Cause and Karel Kuchar's Unmoved Mover ("The Problem of Time In Quantum Geometrodynamics", in "The Arguments of Time", ed. by Jeremy Butterfield, Oxford University Press, Oxford, 1999, p. 193).

I'm afraid you have completely missed the argument in the Buridan donkey paradox.